The chart on the second page of the article used to be located on a Wikipedia page called "Privileged character of 3+1 spacetime". I think it's curious that it seems to be mirrored over a diagonal axis.
That's expected when you realize that time and space are not different things but just two aspects of the same, pseudo-Riemannisn structure that describes our universe. The only thing separating them is a sign which in turn is based purely on convention.
> That's expected when you realize that time and space are not different things [...] The only thing separating them is a sign which in turn is based purely on convention.
The fact that spatial and temporal dimensions require different signs in the metric signature (regardless of whether you take the +--- or the -+++ convention) should suggest that time and space dimensions are indeed, in some sense, different (in the sense of "not interchangeable").
There's a reason space-time is usually modelled as a 4-dimensional Lorentizan manifold, rather than a 4-dimensional Riemannian manifold.
What do you mean with usually? Spacetime (i.e. the object of discussion in relativity) is by definition not positive definite. You said it yourself that space and time require different signs. But they are still interchangeable - without changing physics. On top of that, all that a Lorentz transformation (at least for boosts) does is rotate (hyperbolically) space and time into each other. Seeing space and time as two different things rather than interchangeable components of a more fundamental object is not in line with relativity.
Agree. tx bivectors squares to 1 giving hyperbolic rotations, while xy bivectors squares to -1 giving euclidean rotations.
saying time and space aren't different because spacetime exists sounds like saying electric fields and magnetic fields aren't different because EM field exists.
>like saying electric fields and magnetic fields aren't different because EM field exists.
Funny you should say that, because this idea is well known as S-duality [1] in Quantum Field Theory (and more generalized in String Theory). In fact, if magnetic monopoles (read: charges) exist, you could already see that simply by looking at Maxwell's equations and replacing E->B and B->-E. If you also know a bit about the Maxwell bivector, it's easy to see how closely this duality mirrors how space and time are related.
It is true from SR+Minkowski that space and time can be considered together as a manifold, where (relative motion) boosts rotate smoothly between the dimensions. The signature just decides the interval expression, but doesn't change physics.
But is also true (SR, esp GR) that the important thing about spacetime is the light-cone structure at each spacetime point. Space-like and time-like directions (2 for time-like, past and future) really are different. The forward and backward light cones define a null-surface with zero interval (everywhere, all at once, a la photon).
Relative velocities tilt light cones to bring the local and global views into agreement. Somehow...
However... Proper time is asymmetric, past and future really are different here and now. Local proper time seems to be inexorable, and ~independent of the universe. It is the clock tick we all feel. Proper time seems directed, but non-transformable. We always experience the same rate of 1s/s, looking at our local inertial atomic clocks.
It is confusing to hold all those ideas in my mind at the same time. I have not seen a good explanation that resolves the confusion (I have seen the equations :)
My guess is that local proper time is a real progression, perhaps by axiom, not dependent on cause-effect, or 2nd Law, or QM entanglement, or .. anything else. However, relative time is accurately described by SR, GR and some future QG.
Also, QM is fundamentally wrong, because it is background-dependent. GR showed us that the more profound approach is to include gravity, space, time, matter and energy as dynamic participants in the same framework of laws. QM (Schrodinger/Dirac/QED/QFT...) seems provincial, and provisional, because it assumes a spacetime background - which cannot possibly be true, as is.
I suppose I align with the GR-istas, especially Penrose, who seem to understand this conformal/twistor structure deeply, and use it to lead their intuition for speculative future theories, of cosmology, and also QG.
